# Antiprism

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An antiprism is a polytope built from two parallel bases, dual to one another. The bases are joined to each other by various facets which link corresponding elements together. Antiprisms generalize the concept of a polygonal antiprism to higher (and lower) dimensions. An antiprism is also an alterprism if the base is self-dual.

Under this definition, the octahedron atop cube segmentochoron can be seen as a cubic antiprism or an octahedral antiprism, and the hexadecachoron can be seen a tetrahedral antiprism. Unless their bases are self-dual, as happens with the simplex cases, which produce cross polytopes, and with the 5D icositetrachoric antiprism, these antiprisms are generally not isogonal, and unless their bases are regular, they are not even CRF. However, their simple mathematical definition and their relation to more general laces makes them worthwhile to study.

If one of the bases is not only dualized but also centrally inverted with respect to the other, the antiprism may be called a retroprism or crossed antiprism.

Duals of antiprisms are called antitegums.

## Properties

The rank of an antiprism is always the rank of the original polytope plus one.

The vertex figure of an antiprism is always another antiprism, either based either on some facet of the original polytope, or some facet of its dual. Furthermore, the facets of an antiprism other than the bases are duopyramids of elements in the original polytope and elements in the dual.

Cross-sections of an antiprism along the direction separating its bases result in successive expansions of the original polytope.

## Definition

When dealing with convex polytopes, antiprisms can be easily defined as the convex hull of the vertex set formed by a polytope and its dual in non-intersecting hyperplanes. However, this definition does not generalize to non-convex shapes. Hence, we dedicate the rest of this section to giving a general abstract definition.

A section of an abstract polytope is defined as the set of all elements in the polytope lying between any two specified elements. Sections generalize the concepts of elements and element figures. For our following discussion, we'll allow a section to contain a single element, or even no elements.

The antiprism of an abstract polytope may be defined as the set of all of its sections, including the empty section, reverse-ordered by inclusion. In more formal language, we can represent the antiprism of a polytope (P, ≤) as the set

${\displaystyle \{A/B:A\in P,\ B\in P\}\cup \{\varnothing \}}$

ordered by ⊇, where A/B represents the section between A and B.

In this construction, each section encodes the duopyramid made out from its lowest element and its highest element's dual. To make it concrete, it suffices to map all sections between a vertex and the maximal element to a vertex of the original polytope, and all sections involving the minimal element and a facet to a vertex of the dual polytope.